A176129 -id:A176129 - cp's OEIS frontend

A006335 a(n) = 4^n(3n)!/((n+1)!(2n+1)!).

1, 2, 16, 192, 2816, 46592, 835584, 15876096, 315031552, 6466437120, 136383037440, 2941129850880, 64614360416256, 1442028424527872, 32619677465182208, 746569714888605696, 17262927525017812992, 402801642250415636480, 9474719710174783733760, 224477974671833337692160

Offset: 0

N. J. A. Sloane

Number of planar lattice walks of length 3n starting and ending at (0,0), remaining in the first quadrant and using only NE,W,S steps.
Equals row sums of triangle A140136. - Michel Marcus, Nov 16 2014
Number of linear extensions of the poset V x [n], where V is the 3-element poset with one least element and two incomparable elements: see Kreweras and Niederhausen (1981) and Hopkins and Rubey (2020) references. - Noam Zeilberger, May 28 2020

			G.f. = 1 + 2*x + 16*x^2 + 192*x^3 + 2816*x^4+ 46592*x^5 + 835584*x^6 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..700
Andrei Asinowski, Cyril Banderier, and Sarah J. Selkirk, From Kreweras to Gessel: A walk through patterns in the quarter plane, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #30.
Olivier Bernardi, Bijective counting of Kreweras walks and loopless triangulations, Journal of Combinatorial Theory, Series A 114:5 (2007), 931-956.
M. Bousquet-Mélou, Walks in the quarter plane: Kreweras' algebraic model, arXiv:math/0401067 [math.CO], 2004-2006.
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008.
Sam Hopkins and Martin Rubey, Promotion of Kreweras words, arXiv:2005.14031 [math.CO], 2020.
G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.
G. Kreweras and H. Niederhausen, Solution of an enumerative problem connected with lattice paths, European J. Combin., 2 (1981), 55-60.

Equals 2^(n-1) * A000309(n-1) for n>1.
Cf. A098272. First row of array A098273.
Column of A176129, A214631, A214722, A340591.

[4^n*Factorial(3*n)/(Factorial(n+1)*Factorial(2*n+1)) : n in [0..20]]; // Wesley Ivan Hurt, Nov 16 2014

A006335:=n->4^n*(3*n)!/((n+1)!*(2*n+1)!): seq(A006335(n), n=0..20); # Wesley Ivan Hurt, Nov 16 2014

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, 3 n], {n, 0, 25}] (* Manuel Kauers, Nov 18 2008 *)
Table[(4^n (3 n)! / ((n + 1)! (2 n + 1)!)), {n, 0, 200}] (* Vincenzo Librandi, Nov 17 2014 *)

{a(n) = if( n<0, 0, 4^n * (3*n)! / ((n+1)! * (2*n+1)!))}; /* Michael Somos, Jan 23 2003 */

def a(n):
    return (4**n * binomial(3 * n, 2 * n)) // ((n + 1) * (2 * n + 1))
# F. Chapoton, Jun 01 2020

G.f.: (1/(12*x)) * (hypergeom([ -2/3, -1/3],[1/2],27*x)-1). - Mark van Hoeij, Nov 02 2009
a(n+1) = 6*(3*n+2)*(3*n+1)*a(n)/((2+n)*(2*n+3)). - Robert Israel, Nov 17 2014
a(n) ~ 3^(3*n + 1/2) / (4*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Mar 26 2016
E.g.f.: 2F2(1/3,2/3; 3/2,2; 27*x). - Ilya Gutkovskiy, Jan 25 2017

Edited by N. J. A. Sloane, Dec 20 2008 at the suggestion of R. J. Mathar

A214801 Number of solid standard Young tableaux of shape [[2*n,n],[n]].

Original entry on oeis.org

1, 6, 174, 7020, 325590, 16290708, 854630476, 46305862488, 2568272967270, 144984584562180, 8298621602461476, 480298712286979560, 28052543639835133692, 1650956086756046986440, 97790578929910135588440, 5824509559447044190027952, 348581174512709008160833158

Offset: 0

Alois P. Heinz, Jul 28 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..555
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Central row elements of A214775.

Column k=2 of A176129.

b:= proc(x, y, z) option remember; `if`(z>y, b(x, z, y),
      `if`({x, y, z}={0}, 1, `if`(x>y and x>z, b(x-1, y, z), 0)+
      `if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0)))
    end:
a:= n-> b(2*n, n, n):
seq(a(n), n=0..20);

b[x_, y_, z_] := b[x, y, z] = If[z>y, b[x, z, y], If[Union[{x, y, z}] == {0}, 1, If[x>y && x>z, b[x-1, y, z], 0] + If[y>0, b[x, y-1, z], 0] + If[z>0, b[x, y, z-1], 0]]]; a[n_] := b[2n, n, n]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

Recurrence: (n-1)*n^2*(2*n-1)*(2*n+1)*(4*n-1)*(4*n+1)*(392*n^4 - 2044*n^3 + 4216*n^2 - 3944*n + 1377)*a(n) = 2*(n-1)*(1859648*n^10 - 13670048*n^9 + 43255264*n^8 - 75152192*n^7 + 75863336*n^6 - 41825576*n^5 + 7317576*n^4 + 5067372*n^3 - 3441344*n^2 + 785094*n - 59535)*a(n-1) - 4*(2*n-3)*(4*n-7)*(4*n-5)*(1310848*n^8 - 7998592*n^7 + 19695952*n^6 - 24269488*n^5 + 15125236*n^4 - 3514192*n^3 - 1066614*n^2 + 533457*n - 45927)*a(n-2) + 5184*n*(2*n-5)*(2*n-3)*(4*n-11)*(4*n-9)*(4*n-7)*(4*n-5)*(392*n^4 - 476*n^3 + 436*n^2 - 76*n - 3)*a(n-3). - Vaclav Kotesovec, Aug 31 2014

a(n) ~ sqrt((5*sqrt(5)-11)/4) * 64^n / (Pi*n). - Vaclav Kotesovec, Aug 31 2014

A215686 Number of solid standard Young tableaux of shape [[3*n,n],[n]].

Original entry on oeis.org

1, 12, 690, 52808, 4558410, 420421056, 40433534960, 4002511248720, 404653074076602, 41573640435563720, 4325688482694408060, 454713687334494619200, 48204482093235945250800, 5146506898529612988887424, 552782991828545241240684480, 59682974236253934536767852960

Offset: 0

Alois P. Heinz, Aug 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Column k=3 of A176129.

b:= proc(x, y, z) option remember; `if`(z>y, b(x, z, y), `if`(z>x, 0,
      `if`({x, y, z}={0}, 1, `if`(x>y and x>z, b(x-1, y, z), 0)+
      `if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0))))
    end:
a:= n-> b(3*n, n, n):
seq(a(n), n=0..20);

b[x_, y_, z_] := b[x, y, z] = If[z>y, b[x, z, y], If[Union[{x, y, z}] == {0}, 1, If[x>y && x>z, b[x-1, y, z], 0] + If[y>0, b[x, y-1, z], 0] + If[z>0, b[x, y, z-1], 0]]]; a[n_] := b[3n, n, n]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

Recurrence: 144*n^2*(2*n-1)^2*(3*n-1)*(3*n+1)*(6*n-1)*(6*n+1)*(1054762104*n^8 - 12255641802*n^7 + 61733195125*n^6 - 176088471031*n^5 + 310993546391*n^4 - 348002018003*n^3 + 240698328276*n^2 - 93950633268*n + 15815567520)*a(n) = 3*(16185096657265536*n^16 - 237611988564522912*n^15 + 1583108012261189688*n^14 - 6336788565970939206*n^13 + 16995962918085979601*n^12 - 32239020575894967680*n^11 + 44456039640264512829*n^10 - 45040072648450035120*n^9 + 33379794951068698383*n^8 - 17660683589724361536*n^7 + 6240753949747677391*n^6 - 1170756653568973234*n^5 - 67328664931986180*n^4 + 105177779152514568*n^3 - 28053879047153568*n^2 + 3460226308012800*n - 166227489792000)*a(n-1) - 5*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(1065259096459008*n^12 - 11533800266013504*n^11 + 53860225549987304*n^10 - 142107589766231326*n^9 + 232900286313689643*n^8 - 245074012300774359*n^7 + 164357596249809711*n^6 - 65773579693921743*n^5 + 11793083348968270*n^4 + 1470831449279884*n^3 - 1161186153316104*n^2 + 208213965227520*n - 12225693916800)*a(n-2) + 97200*(5*n-14)*(5*n-13)*(5*n-12)*(5*n-11)*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(1054762104*n^8 - 3817544970*n^7 + 5477041423*n^6 - 3991100299*n^5 + 1435002321*n^4 - 129425495*n^3 - 68369752*n^2 + 19574652*n - 1364688)*a(n-3). - Vaclav Kotesovec, Aug 31 2014

a(n) ~ sqrt(330*sqrt(33)-1770)/36 * 5^(5*n) / (Pi * n * 3^(3*n)). - Vaclav Kotesovec, Aug 31 2014

A246619 Number of solid standard Young tableaux of shape [[4*n,n],[n]].

Original entry on oeis.org

1, 20, 1876, 229680, 31497284, 4600393936, 699440711760, 109341854545792, 17445620031680100, 2827280025640259280, 463882742476664594512, 76875122571167921990080, 12845419277094419018993808, 2161338658294952555703260480, 365816910931667192749720139072

Offset: 0

Vaclav Kotesovec, Aug 31 2014

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Cf. column k=4 of A176129.

Cf. A214801, A215686.

a(n) ~ sqrt(4*sqrt(3)-6) * 3^(6*n+1) / (Pi * n * 2^(2*n+3)). - Vaclav Kotesovec, Aug 31 2014

A215123 Number of solid standard Young tableaux of shape [[n^2,n],[n]].

Original entry on oeis.org

1, 2, 174, 52808, 31497284, 31113230148, 46190668836656, 96484621769643360, 270280816277448460968, 979042561410295182717884, 4456728497956906393963534248, 24916868994347706845906490576432, 167903137478620963997932010166057408

Offset: 0

Alois P. Heinz, Aug 03 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Central row elements of A215122.

Main diagonal of A176129.

b:= proc(x, y, z) option remember; `if`(zy and x>z, b(x-1, y, z), 0)+
      `if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0)))
    end:
a:= n-> b(n^2, n, n):
seq(a(n), n=0..15);

$RecursionLimit = 1000; b[x_, y_, z_] :=  b[x, y, z] = If[zy && x>z, b[x-1, y, z], 0] + If[y>0, b[x, y-1, z], 0] + If[z>0, b[x, y, z-1], 0]]]; a[n_] := b[n^2, n, n]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

a(n) ~ exp(2*n+2) * n^(2*n-1) / (2*Pi). - Vaclav Kotesovec, Jan 19 2015

A246620 Number of solid standard Young tableaux of shape [[5*n,n],[n]].

Original entry on oeis.org

1, 30, 4140, 738192, 146955276, 31113230148, 6851807953900, 1550766110966400, 358116337203378732, 83984165552626389864, 19937272615715693766528, 4779986445560522545646400, 1155414579663560935856564700, 281212253617692376239817669056

Offset: 0

Vaclav Kotesovec, Aug 31 2014

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..150
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Cf. Column k=5 of A176129.

Cf. A214801, A215686, A246619.

a(n) ~ sqrt(7/10*(13*sqrt(65)-83))/10 * 7^(7*n) / (Pi * n * 5^(5*n)). - Vaclav Kotesovec, Aug 31 2014

A215687 Number of solid standard Young tableaux of shape [[2*n,2],[2]].

Original entry on oeis.org

0, 16, 174, 690, 1876, 4140, 7986, 14014, 22920, 35496, 52630, 75306, 104604, 141700, 187866, 244470, 312976, 394944, 492030, 605986, 738660, 891996, 1068034, 1268910, 1496856, 1754200, 2043366, 2366874, 2727340, 3127476, 3570090, 4058086, 4594464, 5182320

Offset: 0

Alois P. Heinz, Aug 20 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Row n=2 of A176129.

a:= n-> (-7+(7+(12+4*n)*n)*n)*n;
seq(a(n), n=0..40);

LinearRecurrence[{5,-10,10,-5,1},{0,16,174,690,1876},40] (* Harvey P. Dale, Jul 08 2017 *)

G.f.: 2*(3*x^3-10*x^2+47*x+8)*x/(1-x)^5.

a(n) = n*(2*n-1)*(2*n^2+7*n+7).

A215688 Number of solid standard Young tableaux of shape [[3*n,3],[3]].

Original entry on oeis.org

0, 192, 7020, 52808, 229680, 738192, 1950512, 4488180, 9314448, 17841200, 32050452, 54630432, 89126240, 140105088, 213336120, 315984812, 456821952, 646447200, 897527228, 1225048440, 1646584272, 2182577072, 2856634560, 3695840868, 4731082160, 5997386832

Offset: 0

Alois P. Heinz, Aug 20 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Row n=3 of A176129.

a:= n-> `if`(n=0, 0, 32+(-312+(-362+(171+(657+(405+81*n)*n)*n)*n)*n)*n/4):
seq(a(n), n=0..40);

G.f.: 4*(8*x^6-41*x^5+105*x^4+181*x^3+1925*x^2+1419*x+48)*x/(1-x)^7.

a(n) = (3*n-1)*(3*n-2)*(3*n^2+13*n+16)*(3*n^2+5*n+4)/4 for n>0, a(0) = 0.

A246621 Number of solid standard Young tableaux of shape [[6*n,n],[n]].

Original entry on oeis.org

1, 42, 7986, 1950512, 530931786, 153580152492, 46190668836656, 14274134610246720, 4500027052542851130, 1440557297650459814996, 466776334221187994469180, 152741149363060061495819904, 50388989722150284436348268528, 16737346518387797143628281698720

Offset: 0

Vaclav Kotesovec, Aug 31 2014

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..150
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Cf. column k=6 of A176129.

Cf. A214801, A215686, A246619, A246620.

a(n) ~ sqrt((7*sqrt(21)-23)/6)/3 * 8^(8*n) / (Pi * n * 6^(6*n)). - Vaclav Kotesovec, Aug 31 2014

A246632 Number of solid standard Young tableaux of shape [[7*n,n],[n]].

Original entry on oeis.org

1, 56, 14014, 4488180, 1600210950, 606021260768, 238569956361984, 96484621769643360, 39804374977976438550, 16673506088758905946560, 7069147910017295001150416, 3026650659542070433624365472, 1306409837754633649293171595680, 567753549048151735754482750312560

Offset: 0

Vaclav Kotesovec, Aug 31 2014

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..150
Wikipedia, Young tableau

Cf. column k=7 of A176129.

a(n) ~ 3/14*sqrt(3/14*(5*sqrt(105)-33)) * 9^(9*n) / (Pi * n * 7^(7*n)). - Vaclav Kotesovec, Aug 31 2014

cp's OEIS Frontend

A006335 a(n) = 4^n*(3*n)!/((n+1)!*(2*n+1)!).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A214801 Number of solid standard Young tableaux of shape [[2*n,n],[n]].

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A215686 Number of solid standard Young tableaux of shape [[3*n,n],[n]].

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A246619 Number of solid standard Young tableaux of shape [[4*n,n],[n]].

Views

Author

Keywords

Links

Crossrefs

Formula

A215123 Number of solid standard Young tableaux of shape [[n^2,n],[n]].

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A246620 Number of solid standard Young tableaux of shape [[5*n,n],[n]].

Views

Author

Keywords

Links

Crossrefs

Formula

A215687 Number of solid standard Young tableaux of shape [[2*n,2],[2]].

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A215688 Number of solid standard Young tableaux of shape [[3*n,3],[3]].

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A006335 a(n) = 4^n(3n)!/((n+1)!(2n+1)!).